American Journal of Operations Research, 2013, 3, 439-443
http://dx.doi.org/10.4236/ajor.2013.35042 Published Online September 2013 (http://www.scirp.org/journal/ajor)
Scheduling Jobs with a Common Due Date via
Cooperative Game Theory
Irinel Dragan
University of Texas at Arlington, Mathematics, Arlington, USA
Email: [email protected]
Received October 18, 2012; revised December 30, 2012; accepted January 7, 2013
Copyright © 2013 Irinel Dragan. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Efficient values from Game Theory are used, in order to find out a fair allocation for a scheduling game associated with
the problem of scheduling jobs with a common due date. A four person game illustrates the basic ideas and the computational difficulties.
Keywords: Schedule; Efficient Value; Egalitarian Value; Egalitarian Nonseparable Contribution; Shapley Value; Cost
Excesses; Lexicographic Ordering; Cost Least Square Prenucleolus
1. A Scheduling Game and Simple Solutions
A machine may process n jobs, J1 , J 2 , , J n , with
the completion times p1 , p2 , , pn , all positive numbers.
No two jobs can be simultaneously done, and for all jobs
there is a common due date d positive. Any schedule
 is a sequence of jobs, and no preemption is allowed.
The schedule is determined by the numbers
Ci   , i  N , the completion dates of the jobs J i in
 . Any deviation from the due date will be penalized,
either an early or a late completion relative to the due
date. The total time deviation in a schedule  is given
by
     Ci    d .
(1)
iN
The usual scheduling problem is to: find out the
schedule  * for which the total deviation is minimal.
In [1], J. J. Kanet solved the problem for the case when
the sum of completion times is smaller than, or equal to
d , and gave an algorithm for computing a schedule with
a minimal deviation. Of course, this algorithm may be
used to find the total penalty for this schedule and also to
find the total penalty for any minimal deviation corresponding to any subset of jobs. This makes sense in the
case when the costs of the deviations, early or late, are
proportional to their size. In the following, we assume
that the costs are equal to the penalties. A more general
case other than Kanet’s has been solved by M. U. Ahmed
and P. S. Sundararaghavan in [2]. In [3], N. G. Hall and
M. E. Posner consider similar problems. The literature
Copyright © 2013 SciRes.
connected to more general cases is huge, and the conclusions obtained in the present paper can be applied to
most other cases. For the present discussion, the simplest
case offered by Kanet’s algorithm, with the above assumptions, is good enough to suggest similar approaches
in all other cases, in connection with a new problem to be
introduced below.
Assuming that the grand coalition has been formed
and the total penalty for early and late deviation, w  N  ,
has been computed by some algorithm, a new problem is:
how much should be a fair individual penalty for each
job?
To answer the question, we now build the following
cooperative game with transferable utilities: let
N  1, 2, , n be the set of players, the player i be
the customer ordering the job J i , for each
i  1, 2, , n Consider any coalition of customers,
S , S  N , S  , and notice that if S  i , then the
minimal schedule starts the corresponding job at d  pi ,
and there will be no deviation from the due date. Therefore, if we denote the deviation for coalition S , by
w  S  , we have w i   0. An algorithm for computing the minimal deviation, for example Kanet’s algorithm, will provide the total deviation w  S   0, when
S  2. In this way, we get a cooperative TU game
 N , w , in which we want to divide fairly w  N  .
To make the paper self contained let us sketch Kanet’s
algorithm which will be used in the example shown beAJOR
I. DRAGAN
440
low. Let S be any coalition and denote by BS , (before S ), a sequence of jobs which were already selected,
with non-increasing processing times and the last job
ending at d ; also, denote by AS , (after S ), another
sequence of jobs which were already selected, with nondecreasing processing times and starting at d . Now,
assume that we have BS  AS  , BS  AS , Kanet’s
algorithm is continuing to build the partition of S as
follows: if BS  AS  S  1, select the non-selected
element of S with a maximal processing time and take
it as the last job in BS , (now we have
BS  AS  1 ). Further, if with the new BS , we have
BS  AS  S  1, then choose the non-selected job in
S with a maximal processing time and take it as the
first job in AS , (that is starting at d ). Repeat the procedure, selecting alternatively players in BS , then in
AS , until S is exhausted; then, the time deviation is
computed by formula (1) for the corresponding schedule
and in the same way for any subset of players.
Example 1: Let  J1 , J 2 , J 3 , J 4  be a set of jobs to be
processed on a single machine, with the processing times
p1  12, p2  10, p3  8, p4  5, and the due date is
d  39, such that the Kanet’s condition shown above
holds. We can compute for the set of players
N  1, 2,3, 4 , and its subsets, the total penalties. Kanet’s algorithm will generate the game
w 1   w 2   w 3   w 4   0,
w 1, 2   10, w 1,3   w 2,3   8,
w 1, 4   w 2, 4   w 3, 4   5,
w 1, 2,3   18, w 1, 2, 4   15,
w 1,3, 4   w 2,3, 4   13, w 1, 2,3, 4   28.
This corresponds to the total deviations of all coalitions, and our problem is to: find out how we should divide fairly w  N   28 among the players? We start by
showing two simple solutions: the Egalitarian allocation
and the Egalitarian non-separable contribution allocation.
Denoting the first by x*, we get x*   7, 7, 7, 7  .
Denoting the second by y*, which is given in general by formula
yi *  w  N   w  N  i 

1
  w  N     w  N   w  N   j    , i  N ,
n
j N

(2)
we get the marginal contributions
M j  w  N   w  N   j  , j  N , equal to 15, 15, 13,
10, so that the sum makes 53, and from each marginal
contribution we should subtract 25/4, to obtain
 35 35 27 15 
y*   , , ,  .
 4 4 4 4
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Looking at the characteristic values of our game,
shown in example 1, we see that the players 1 and 2 seem
to be equal, while players 3 and 4 are weaker, hence the
last two should be asked to pay smaller individual penalties.
The first solution does not seem to show this, while the
second seems more fair, we shall see a method below to
compare the fairness of the solutions.
2. Individual Penalties: Set Solutions,
Shapley Value
To evaluate the fairness of a possible solution z , we
may use the excess functions; however, here it seems
more appropriate to use some similar functions that we
shall call the “cost excess” functions. For any coalition
S , S  N , S  , and any allocation z , the cost excess
function is
  S , z    zi  w  S  .
(3)
iS
These are the negatives of usual excess functions, obviously, we have 2n  2 such functions, because for the
grand coalition, for any allocation, by definition we have
  N , z   0. In words, the cost excess is the difference
between what the coalition S will pay in z to contribute as close as possible to the total penalty for herself,
while it is also contributing to the total penalty for N .
Then, what we want to do is to choose the allocation z
which minimizes all cost excesses on the set of allocations. Note that the sum of all cost excesses is a constant,
because for any allocation z we have
   S , z   2n 1 w  N    w  S .
SN
(4)
SN
Moreover, we can define the average cost excess
  w 
1
   S , z ,
2n  1 S  N
(5)
which by formula (4) does not depend on the allocation
z This means that if the allocation of some coalition is
increased, then the allocation of at least one other coalition will be decreased. How the cost excesses are used to
compare the fairness of two allocations is illustrated below.
Example 2: Return to example 1, and write the cost
excesses for that game  N , w  , and any allocation
 1 , z   z 1 ,  2 , z   z 2 ,
 3 , z   z 3 ,  4 , z   z 4 ,
 1, 2 , z   z 1  z 2  10,
 1,3 , z   z 1  z 3  8,
AJOR
I. DRAGAN
 1, 4 , z   z 1  z 4  5,  2,3 , z   z 2  z 3  8,
 2, 4 , z   z 2  z 4  5,  3, 4 , z   z 3  z 4  5,
 1, 2,3 , z   z 1  z2  z 3  18,
 1, 2, 4 , z   z 1  z 2  z 4  15,
 1,3, 4 , z   z 1  z 3  z 4  13,
 2,3, 4 , z   z 2  z 3  z 4  13.
Our problem is to minimize all cost excesses, while we
are on the set of allocations, that is the efficiency condition holds. In other words, we want to minimize the
maximal cost excess, subject to efficiency condition, or
to use another method to solve a multi objective linear
programming problem.
Let us try to evaluate the fairness of the two allocations offered until now.
For the Egalitarian solution, we can compute the cost
excesses and put them in a vector of non increasing excesses, which may be called the vector of unhappiness, as
the components are taken in the order of non increasing
unhappiness
  x *   9,9,9,8,8, 7, 7, 7, 7, 6, 6, 6, 4, 3 .
It follows that the most unhappy coalitions are the two
person coalitions in which one of the players is player
4. For the Egalitarian non separable contribution, we
find the vector of unhappiness
  y * 
 35 35 15 15 15 15 15 27 25 25 25 25 11 15 
 , , , , , , , , , , , , , ,
 4 4 2 2 2 2 2 4 4 4 4 4 2 4
and the most unhappy coalitions are 1 and 2 .
Moreover, we have also 1  x * larger than 1  y * ,
that is the most unhappy coalition in x * is more unhappy than the most unhappy coalition in y *. We can
say that y *. is better than x*, or more fair. Note that
the same thing could be said if some pairs of corresponding components in the two unhappiness vectors are
equal, but the first one which is different is smaller in
  y * than in   x * . In this case, we also write
  y *  L   x * , where L means the lexicographic
order, and read y * is better than x *. Until now, we
have seen two simple solutions belonging to Game Theory, they are one point solutions because each one is providing a unique solution.
One of the set solutions from Game Theory is the
CORE, which for a cost game  N , w  like ours is the
set
CO  N , w 


 z  R :   N , z   0,   S , z   0, S  N , S   .
n
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(6)
441
Any element of the CORE is considered as a good allocation, because such an allocation covers the total penalty for each coalition. Looking at the two simple solutions of example 1, which as seen in example 2 have all
excesses non negative, we see that both are in the CORE,
but, of course, there are others with the same property.
Moreover, we can also see that the sum of excesses
equals 96, that is the worth obtained in formula (4) for
n  4.
The most famous one point solution, which may also
be in the CORE, is the Shapley Value, introduced in [4],
and defined by a set of axioms, describing some basic
properties required for a fair solution. The Shapley Value
was proved there to be given by the formula
SH i  N , w  

 s  1! n  s ! 
S :iS
n!
 v  S   v  S  i   , i  N .
(7)
Example 3: For the game considered in example 1, we
get
SH  N , w    8,8,7,5  .
Computing the cost excesses and ordering them, we
obtain
  SH    8,8,8,8,7, 7,7,7, 7,7,6,6,5,5  .
We get   SH   L   y *  L   x * , because the first
components of the unhappiness vectors are in this order,
hence the Shapley Value is better than the Egalitarian
non separable contribution, which is better than the Egalitarian solution. Note that this may not be the case for
other games. Note also that if the game is large, then the
Shapley Value may not be easy to compute. An algorithm based upon the so called Average per capita formula, given by the author in [5], may be used, as it will
be explained in the next section and the algorithm is allowing even a parallel computation of the Shapley Value.
Similar situations may occur in connection with the other
values.
3. The Cost Least Square Prenucleolus
In the following, we may consider as a solution the Least
Square Prenucleolus of the game, introduced by L. Ruiz,
F. Valenciano and J. Zarzuelo in [6]. This is similar to
the Prenucleolus, introduced in connection with the Nucleolus, due to D. Schmeidler [7], except that this was
defined by means of the following quadratic programming problem
Minimize f  w, z  
   S , z     w
2
,
(8)
SN
subject to
  N , z   0.
(9)
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I. DRAGAN
442
By using the Kuhn-Tucker conditions, in (8), (9), it
has been shown that our problem has a unique solution,
that the authors called the Least Square Prenucleolus,
namely
LSi  N , w  
w N 
n


1 
 nai  w    a j  w   , i  N ,
jN

(10)
n 
where
ai  w  
n 1
 n  2
s 1

 w  S , i  N ,    s  1 .
S :iS

(11)
As the cost excesses are replacing the excesses, we
called this value the Cost Least Square Prenucleolus, but
the expressions (10) and (11) are the same even in our
case.
Example 4: Computing by (10) and (11) this solution
for our game, we get
 129 129 113 77 
LS  N , w   
,
,
, .
 16 16 16 16 
The vector of unhappiness is   LS * (see foot-note).
Notice that the sum of cost excesses is also 96. Now,
checking the comparison of the new solution with the
other three solutions, we obtain
SH  N , w   L LS  N , w   L y*  xL *,
that is the Shapley Value seems to be more fair than the
other three values. Of course, the Schmeidler’s Prenucleolus may also be computed; the computational method,
due to A. Kopelowitz [8], is also shown in [9]. However,
this includes a long computation for solving a sequence
of linear optimization problems. The Prenucleolus would
be the best, by the definition of the value.
Another principle may be used to choose the appropriate allocation: for each allocation available, compute the
difference between the cost for the most unhappy coalition and the happiest coalition and choose the allocation
that gives the smallest difference. Such an allocation
would not give a high difference of costs between the
happy and the unhappy coalitions. In our case, we have
d SH  3, d LS 
129 77 13

 , d y*  5, d x*  6.
16 16 4
Notice that by the last principle the four values are ordered in the same way.
4. Conclusions
The technology put together in the present paper applies
to other scheduling problems in which the associated
scheduling game can still be generated by some algorithm. Some of the following remarks may help:
1) The above discussion was illustrated by examples 1,
2, 3, 4 relative to a four person game. If we have n  5
jobs, under the same conditions like above, Kanet’s algorithm still applies when the Kanet’s assumption holds. If
the objective is different, for example to minimize a
weighted combination of deviations relative to a common
due date, then the algorithm by Hall and Posner should
be used to get the scheduling game.
2) As soon as the game is available, the problem of dividing fairly the worth of the grand coalition is the problem of choosing an efficient value from Game Theory,
for which the computation could be done. As all singletons have a zero worth, the Center of the imputation set
[9] becomes the Egalitarian value, which in general is not
fair. The Egalitarian non-separable contribution may be
an alternative allocation. The Shapley Value, which has a
lot of properties derived from the axioms, is the most
preferred by almost all scientists, but for more than ten
players it becomes difficult to compute. For such large
games it may be better to use the formula given by the
author in [5], called the Average per capita formula
ws  wsi
, i  N ,
s
s 1
sn
SH i  N , w   
(12)
where ws is the average worth of coalitions of size s,
and wsi is the average worth of coalitions of size s,
that do not contain player i, with wni  0, i  N . It is
obvious that the task can be performed by n teams, and
each one computes one ratio for one s.
3) In the computation of the Cost Nucleolus by Kopelowitz’ method [8,9] the passage from one LP problem to the next is not described in details in most sources.
The complementary conditions show which cost excesses
should remain constant on all optimal solutions of the
current problem, and should be kept constant in the next
problem. These equations should replace the corresponding inequalities of the current problem and remain satisfied until we meet a problem which has a unique solution.
The solution of the quadratic programming problem seems
easier to compute.
The generalized nucleolus may be used as a solution of
any Multicriteria Linear Programming problem, as shown
by the author in [10], working with a three person game.
The basic idea appeared in a former paper of the author
[11], as well as in the more recent paper by E. Marchi
and J. A. Oviedo [12].
 129 129 63 63 57 57 113 111 111 55 49 95 83 77 
,
, , , , ,
,
,
, , , , , .
 16 16 8 8 8 8 16 16 16 8 8 16 16 16 
  LS *  
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AJOR
I. DRAGAN
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AJOR